Respuesta :

mhanifa

Given

  • A quadrilateral ABCD with opposite pairs of sides being congruent.

To prove

  • Opposite sides are parallel.

Solution

We'll prove first the triangles formed by diagonal are congruent then we'll prove the angles on either side of diagonal are congruent.

The steps are:

  • Step  ⇒   Statement  ⇒   Reason

=============================================================

  • 1   ⇒  AB ≅ CD and BC ≅ AD     ⇒     Given
  • 2  ⇒  AC = CA                              ⇒    Common side
  • 3  ⇒  ΔABC ≅ ΔCDA                   ⇒    Side-side-side congruence
  • 4  ⇒  ∠BAC ≅ ∠DCA                   ⇒    Corresponding angles of congruent triangles
  • 5  ⇒  ∠BAC and ∠DCA are alternate interior angles  ⇒  Definition of alternate interior angles
  • 6  ⇒  AB║CD                               ⇒   The converse of alternate interior angles theorem
  • 7  ⇒  ∠BCA ≅ ∠DAC                   ⇒   Corresponding angles of congruent triangles
  • 8  ⇒  ∠BCA & ∠DAC are alternate interior angles  ⇒  Definition of alternate interior angles
  • 9  ⇒ BC║AD                                ⇒   The converse of alternate interior angles theorem

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